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Answer :
Final Answer:
The sum of the n terms in each AP is n(n + 2).
Explanation:
The general formula for the sum of an arithmetic progression (AP) with n terms, first term (a), and common difference (d) is:
S = n/2 * [2a + (n - 1) * d]
For the three APs:
AP 1: a = 1, d = 1
AP 2: a = 1, d = 2
AP 3: a = 1, d = 3
Therefore, the sums S1, S2, and S3 are:
S1 = n/2 * [2(1) + (n - 1) * 1] = n(n + 1) / 2
S2 = n/2 * [2(1) + (n - 1) * 2] = n(n + 2) / 2
S3 = n/2 * [2(1) + (n - 1) * 3] = n(n + 5) / 2
Since the question asks for the sum of n terms in each AP, we are essentially interested in the first terms (a) of each progression. These are:
AP 1: 1
AP 2: 1
AP 3: 1
Therefore, the answer is n(n + 2), which applies to all three APs as their first terms are the same (1) and their common differences differ by a constant (1).
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